April 19, 1877] 



NA TURU 



533 



absolutely resembled his parent, the number of children 



who deviated x 



would vary as e 



X e 



or as 



e •■■' '■'' '. Hence the deviations of the children in 

 their amount and frequency would conform to the law, 

 and the modulus of the population of children in the sup- 

 posed case of absolute resemblance to their parents, 

 which we will write C3, is such that — 



We may, however, consider the parents to be multiplied 



Fh; 6. 



and the productivity of each of them to be uniform. h 

 is more convenient than the converse supposition and i' 

 comes to the same thing. So we will suppose the reverted 

 parentages to be more numerous but equally prolific, in 

 which case their modulus will be c^, as above. 



4. Family variability was shown by experiment to 

 follow the law of deviation, its modulus, which we will 

 write V, being the same for all classes. Therefore the 

 amount of deviation of any one of the offspring from the 

 mean of his race is due to the combination of two influ- 

 ences, the deviation of his " reverted " parentage and his 

 own family variability ; both of which follow the law o'' 

 deviation. This is obviously an instance of the well- 

 known law of the " sum of two fallible measures " (Air> , 

 " Theory of Errors," § 43). Therefore the modulus of 

 the population in the present stage, which we will write 

 c^, is equal to x {v- + c^~). 



5. Natural selection follows, as has been explained, the 

 same general law as productiveness. Let its modulus 

 be written s ; then the percentage of survivals among 

 children, who deviate x from the mean, varies as 



e ^'^ , and for the same reasons as those already given, 

 its effect will be to leave the population still in conformity 



Fig. 



Tr^l^^^T^- 



4. c, = f^\v^^c,^ \ 



S. c, 



with the law of deviation, but with an altered modulus, 

 which we will write f,, and 



Putting these together we have, starting with the origi- 

 nal population having a modulus = c^ : — 

 I 



2. fa = rc^' 



3- ^3.= y , VYj^rl 



- V! 



i'C, 



And lastly, as the condition of maintenance of statistical 

 resemblance in consecutive generations : — 



6. c-a = t"o. 



Hence, given the coefficient r and the moduli v,fy s, 

 the value of ^o (or c^) can be easily calculated. 



In the case of simple descent, which wis the one first 



considered, we have nothing to do with c^, but begin from 

 Ci. Again, as both fertility and natural selection are in 

 this case uniform, the values of/ and s are infinite. Con- 

 sequently our equations are reduced to — 



^2 = rcj ; C4 = V [ v" + Ci- \; Ci = Ci, 



whence ^ > _ v'^ 



CA/iL FRIED RICH GAUSS 



Born April 30, 1777, Died February 23, 1855.^ 



r^E MORGAN in his " Budget of Paradoxes " (p. 187), tells 

 the following story : — The late Francis Baily wrote a sin- 

 gular book, ''Account of the Rev. John Flamsieed, the 6rst 

 Astronomer- Royal : " it was published by the Admiralty for 

 distribution, and the author drew up the distribution list. 



' We adopt the date given by the Baron Sartorius von Waltershaiise i in 

 his " Gauss. Zuin Gediichtniss," Leipzig. 1856. Kncyclopedists and other 

 authorities are pretty equally divided between this date and April 23. All 

 the English Cyclopa;diis»e have consulted, with the exception of Chambers's 

 ('874). give April 23. We may also mention that on the list of ^tudents at 

 the Collegium ("arolinum ilie name is Johann Fiiedtich Karl Gauss. We 

 have followed Gauss himself in cur heading. 



